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CMLab

Some basic Applications of Digital filters :

• Least-square fitting of polynomials

Given M data points (tm, um), m=1,2,…,M wish to fit (approximate) these data, in some sense, by a polynomial u=u(t) of degree N where M N+1≧

Principle of least square :

the sum of the squares of the residuals is the least

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CMLab

Ex: consider a set of 5 equally spaced data points.

tm=m for m=-2, -1, 0, 1, 2

um unspecified

fit the above data by a straight line u=A+Bt

in least-square sense.

1 1, 1, 1, 1,5

1 : windowrunning

average moving : 5

1

equation normal

0,

0,

,min

2

2

2

2

2

mmnn

mm

uu

B

BAFA

BAF

BmAuBAF

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CMLab

From the frequency point of view :

S’pose the input function is one of the eigenfunctions in the complex form eiwt of frequency w.

Since the system is linear, the same function will emerge at the output except that it is multiplied by its eigenvalue H(w).

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CMLab

2sin5

25sin

2cos2cos215

1

15

1 22

w

w

ww

eeeewH iwiwiwiw

2

212

1

sin

sin

12

1

...12

1

w

N

iNwwNiiNw

w

N

eeeN

wH

In general :

: transfer function

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CMLab

the more terms used , the more rapid are the wiggles of H(w), and the more the envelope of the wiggles is squeezed toward the frequency axis.

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CMLab

• Least-squares Quadratics and Quartics

Instead of a straight line, fit by a quardratic (or a cubic) u(t)=A+Bt+Ct2

36,9,44,69,84,89,84,69,44,9,3611

21,14,39,54,59,54,39,14,219

2,3,6,7,6,3,27

3,12,17,12,35

,,min

4291

2311

211

351

22

m

m

m

m

CmBmAuCBAF m

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CMLab

• The effect of the higher-degree polynomial is a higher degree of tangency at w=0

• The use of more terms in the smoothing formula makes the curve come down sooner.

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CMLab

• Modified least squares :

Smoothing Window for 2N+1 points

modified Smoothing Windows for 2N+1 points

: reduce the two end values to one-half their assigned values

22

2

212

cossin

sin

2

1

cossin

sin

2

1

ww

w

N

Nw

N

wNw

NwH

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CMLab

• The curve will come down more rapidly

• The main lobe is slightly wider.

★ the end constants can be chosen as a parameter

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CMLab

• Differences and Derivatives

The difference operator

the operator annihilates a polynomial Pn(x) of degree n in X; that is

nKn

K

nnn

uu

uuu1

1

1n

01 xPnn

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CMLab

from frequency point of view :

iwtKwiKwKiwtK

iwtwiw

iwtiw

iwttiwiwt

eeie

eie

ee

eee

22

22

1

sin2

sin2

1

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CMLab

Since , so the amplification at frequency w is contained in the factor

; decreases the amplitude of any frequency△ ; there is an amplification

12 iKw

eiK

Kw2sin2

w

w

3

30

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high-pass behavior of the difference operator Δk

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CMLab

Differences are also used to approximate derivatives.• Central difference formula

from the formula (with h=1)

iwtiwt

nnn

iwetuetu

h

uuu

so , set

...(1) 2

11

wiee iwiw sin2

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CMLab

The ratio of the calculated to the true answer (which is iw) is :

w=0, R=1

w0, |R|<1 the formula underestimates the value of the derivatives for all other freqs.

For the 2nd derivate :

w

w

iw

wiR

sin

2

sin2

true

calculated

2

2

2sin

121

w

w

R

tutututu

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CMLab

Spencer’s smoothing formula:

15-point:

21-point:

,...57,60,57,47,33,18,6,2,5,5,3,1

3,6,5,3,21,46,67,74,67,46,21,3,5,6,3

3501

3201

not informative

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CMLab

plot the logs of the numbers |H(w)|

20 log|ratio| = decibel units (dB)

20 dB = factor of 10

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CMLab

• Missing Data and Interpolation :The reasons or situations for the occurrence of “missin

g data” in a long record of data :• the measurements may never have been made;

they may have been misrecorded and thus later removed

• the formula used to compute successive values of the function may have involved an indeterminate form, such as at x=0, and the computer refused to divide by zero.

xxsin

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CMLab

Usually, an interpolation formula based on the assumption that the data locally is a polynomial of some odd degree.

This is equivalent to the assumption that the next higher-order difference is zero.

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CMLab

For instance, k=4

note: the sum of the sequences of the binomial coeffs. of order N is . Hence for the 2k th difference formula, the noise amplification is

211261

11224

44

0464

nnnnn

nnnnn

uuuuu

uuuuu

NNC2

1222

2242

kC

CCN

kk

kk

kk

A

31.1,3

,2

5.0,1

1817

A

A

A

Nk

Nk

Nk

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CMLab

If set

then wwwH

eu iwnn

2coscos431

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H(0)=1 However, for high freqs, the value is not too

good, particularly for very high freqs.• Negative values on the graph of the transfer functi

on imply a change in sign.

The above figure points out the damager of interpolating a missing value when the data is noisy, which means that the data has numerous high frequencies.

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CMLab

Interpolation midpoint values : linear interpolation gives

if 4 adjacent points are used

221 cos

21

21

wnnn wHuuu

wwH

uuuuu

w

nnnnn

23

281

161

coscos9

9923

21

21

23

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CMLab

• A Class of Nonre cursive Smoothing Filters

Design of filters :

H(0)=1 exact at dc(lowest freq.)

H()=0 no highest freq. get through

cwbwawH

eu

aubucubuauyiwn

n

nnnnnn

cos22cos2

let

2112

L.P.

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CMLab

Since H()=0 the factor [cos w+1] had to occur

Now one can select a filter that approximately meets the requirement.

1coscos14

2022

122

81

21

41

awwawH

ac

b

cba

cba

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CMLab

for Ex.1, if we require

aaH 221 21

21

2cos1cos

262

21

211281

wwwH

uuuuuy nnnnnn

Ex.2

if we set

21

21

2

1141

21

22

cos

2

wwH

uuuy

H

nnnn

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CMLab

Ex. 3. S’pose that we try to do as well as possible in the neighborhood of zero freq.

21

0

23

2

0

3

3

80

0

10

adw

wHd

dw

wdH

H

w

w

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CMLab

Ex: we pick our filter form as

3cos1cos

4104

41

2112161

wwwH

uuuuuy nnnnnn

1121

21

21

11

and

0,1 and

2cos2;

31

61

nnnn

ff

nnnn

uuuy

ba

fHfH

bfawHaubuauy

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CMLab

• Integration : Recursive Filters

Trapezoid Rule (using y0=0)

2

221

0

121

1

sin2

cos

1

1 where

form theof is

0 , Set

w

w

iw

iw

itw

iwt

nnnn

ie

ewA

ewAty

yetu

uuyy

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CMLab

The true answer for integration of is iwte iwtiw e1

...1

sincos

true

calculated

72012

2

221

42

ww

w

ww

iww

wAR

0 ,

1 , 0At

21

Rfw

Rw

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CMLab

Simpson’s Rule :

w=0 , R=1

and has a tangency through the 3rd derivative

1131

11 4 nnnnn uuuyy

1801

cos2

3

4

4

sin31

wwwHR

ee

eewH

ww

iw

w

iwiw

iwiw

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CMLab

• Midpoint integration formula (using y0=0)

2

2

1

sin

21

w

w

nnn

R

uyy

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CMLab

• Leo Tick Formula

1111 3584.02832.13584.0 nnnnn uuuhyy

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CMLab

• Simpson’s formula amplifies the upper part of Nyquist interval (the higher frequencies) where as the trapezoid rule damps them out.

• In the presence of noise. Simpson’s formula is more dangerous to use than are the trapezoid or midpoint formulas. But when there is relatively little high freq. In the function being integrated, then the flatness of Simpson’s formula for low freqs. show why it is superior.